Probability of Meeting Within a 15-Minute Window

This interview question considers two people who independently arrive during a fixed time window, each willing to wait only a limited amount of time for the other. The key condition is that they meet if and only if their arrival times are sufficiently close, but they are also constrained by the start and end of the overall window. The setup is a classic example of a continuous-time arrival problem and often appears in quant trading, market-making, and data science interviews to test comfort with probabilistic reasoning in continuous settings and with interpreting conditions as regions in a plane.

The solution leans on modeling the arrival times as independent continuous random variables with a uniform distribution over an interval. Interviewers expect candidates to translate the timing and waiting rules into inequalities relating the two arrival times, then visualize these as a region in a square representing the joint distribution. It probes understanding of joint densities, geometric probability via area ratios, symmetry arguments, and careful handling of boundary effects. Strong answers also communicate assumptions clearly and check that the resulting probability is intuitively reasonable.

What it tests

When two independent random variables are uniformly distributed over an interval, the joint distribution of their outcomes can be represented as a square (or hypercube in higher dimensions), and constraints on their relationship (like being within a certain distance) carve out geometric regions within that square. The probability of a certain event (such as the two arrivals being within 15 minutes) is then the ratio of the area (or volume) of the region satisfying the constraint to the total area. This geometric approach works because uniform distributions make all points in the square equally likely, so probabilities reduce to ratios of areas. The boundaries of the region are determined by the constraint's algebraic form, which often translates to lines or curves in the plane. This method generalizes to any problem where the event is defined by a simple relationship between two or more uniformly random variables.